Let $\big( X_t , t\geq 0 \big)$ be a measurable process, that is, $$\big( t, \omega \big) \in \mathbb{R}_+\times\Omega\longmapsto X_t(\omega)\in\mathbb{R} \quad\text{is $\mathscr{B}\big( \mathbb{R}_+ \big)\otimes\mathscr{F} \big/\mathscr{B}\big(\mathbb{R} \big)$-measurable. } $$ $T : \Omega\to \big[ 0, +\infty \big]$ be a random time.
Show that the collection $\mathscr{G}$ of all sets of the form $\big( X_T\in A \big)$ and $\big( X_T\in A \big)\cup \big( T = \infty \big)$, $A\in\mathscr{B}(\mathbb{R} )$ is a $\sigma$-algebra on $\Omega$.
Problem 1.17. Karatzas et Shreve.
My confusion is how to understand this question given that $X_T$ is undefined on $\big( T = +\infty \big)$.
EDIT: there would be no problem if we make a change as follows:
Show that the collection $\mathscr{G}$ of all sets of the form $\big( X_T\in A , T < +\infty \big)$ and $\big( X_T\in A , T < +\infty\big)\cup \big( T = \infty \big)$, $A\in\mathscr{B}(\mathbb{R} )$ is a $\sigma$-algebra on $\Omega$.
